in this video we are going to review how to find the volume of a solid using cross sections so there's two formulas that you need to be aware of the volume can be found using this equation is the integration from a to b of the area function and if it's in terms of x then the cross sections has to be perpendicular to the x-axis you can also use this equation where c and d are y values if it's in terms of y then the cross sections has to be perpendicular to the y axis so for this problem the cross sections are perpendicular to the x-axis so we're going to use this formula now let's go ahead and draw a graph so y is equal to the square root of x and it's bounded by the x axis and the line x equals four now if we draw a square the cross sections are squares the area of the square is basically side squared and s is perpendicular to the x-axis s represents the base of the cross section so notice that s is the same as y so the area which is s squared is also equal to y squared and y is the square root of x our goal is to get the area function in terms of x so a of x is equal to x so if we integrate it from a to b this will give us the volume so we're going to integrate it from zero to four and a of x is simply x the antiderivative of x is x squared divided by two evaluated from zero to four so first let's plug in four and then we will plug in zero four squared is 16 16 divided by two is eight so that is the answer now another way in which you can graph this particular problem is you can draw it like this if you want to see why we set it up the way we did so this is the graph y equals square root x and we said this that line is basically s it represents the base of the square and let's draw the square so there is a square you can draw that way too and so this is s as well and so you can see that the area is s squared and you can also see that y is equal to s so you can graph it that way it may help you to visualize it better number two find the volume of the solid bounded by the x-axis the y-axis and the line y equals four minus x over 2 using cross sections of semi-circles that are perpendicular to the x-axis so if the cross-sections are perpendicular to the x-axis ultimately we need to use this equation to find the volume so we got to find the area of the cross section in terms of x but now let's go ahead and graph the equation so we have a line which is four minus x over two we could rewrite it as negative one over two x plus four so now it's in a slope intercept form so the slope is negative one-half and the y-intercept is four so we could graph it at the point zero four and if the slope is negative it's a decreasing function now let's find the x-intercept to do that replace y with zero and solve so we can multiply both sides by two negative half times two is negative one so this is going to be negative x four times two is eight so if we add x to both sides we see that x is equal to eight so the x intercept is eight and the y intercept is four now the cross sections are perpendicular to the x axis so that means s is parallel to the y-axis so this is s now we can graph the function another way too let's say that's the x-axis this is the y-axis and the cross sections are perpendicular to the x-axis which means they're parallel it's the y-axis and they're basically a semi-circle so that's the graph and this is s so as you can see s represents the diameter of the semicircle now we know that the area of a semi-circle is one-half pi r squared and so the radius is half of s so s is equal to two r and one half of s is equal to r so let's replace r with s divided by two so the area is one half pi times one half s squared one half squared is basically one fourth if you do one over two times one over two you get one over four so this is one over four s squared so therefore the area is one half times one fourth is one eighth so it's one over eight pi s squared and if you look at the graph we can see that s is the same as y so s equals y so the area in terms of y is one over eight pi y squared now let's get rid of a few things let's make some space so at this point we can replace y with four minus x over two or let's just use let's use that so the area is going to be 1 over 8 pi times 4 minus 1 half x squared so now we have the area function in terms of x which means we can now use this equation to calculate the volume so the graph was bounded between 0 and 8. so a is 0 b is 8 and then we can integrate the area function at this point and let's not forget dx now before we find the integration of that function we need to foil four minus one half x square so that's four minus one half x times itself so four times four that's equal to sixteen and then we have four times one half x which is negative two x and negative half x times four is also negative two x and then negative one half x times negative one half x that's positive one fourth x squared so this becomes 1 4 x squared minus 4 x plus 16 if we combine the middle terms so therefore the volume is the integration from 0 to 8. over eight pi one fourth x squared minus four x plus sixteen d x so now let's take the constant and move it to the front so the volume is 1 8 pi integration from 0 to 8. one over four x squared minus four x plus sixteen in parentheses with a dx out in the front so now let's find the anti-derivative the anti-derivative of 1 4 x squared that's 1 4 x to the third divided by 3. the inside derivative of 4x is going to be 4x squared divided by 2 and for 16 it's going to turn into 16x evaluated from zero to eight and we still have one eighth pi in front so let's plug in eight so four times three is twelve so that's one over twelve times eight to the third power four divided by two is two and then we have eight squared plus sixteen times eight and then minus we plug in zero to everything it's all going to equal zero now what i'm going to do is multiply or i'm going to distribute the 1 over 8 to every term because they all have 8 inside so therefore this 8 will cancel 1 8 out of each term that we see there so that's going to simplify our calculations so then this becomes 1 over 12 and instead of 8 to the third is now 8 squared we have to take off an 8. and this is going to be two times eight plus sixteen and we're gonna have a pie in front now negative two times eight is sixteen plus sixteen those two will cancel so now we just have this term to deal with eight squared is 64. so we have 64 times the pi outside divided by 12. 64 pi over 12 we can reduce it if we divide both numbers by four this is going to be 16 pi divided by three and so that is the answer now if you want to get the decimal value for it if you type in 16 pi divided by 3 in your calculator you should get five 16.755 that's the decimal value so that's the volume of the solid